Physics — Lagrangian & Hamiltonian Mechanics
The Euler–Lagrange equation and Hamilton's equations.
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L = T - V
\frac{d}{dt}\frac{\partial L}{\partial \dot{q}} - \frac{\partial L}{\partial q} = 0
H = T + V
\dot{q} = \frac{\partial H}{\partial p}, \quad \dot{p} = -\frac{\partial H}{\partial q}